4.2 The Dynamic Equation

The Landau-Lifshitz-Gilbert equation (Eq. (3.8)) is a system of ordinary differential equations (ODEs), which can be written in a general form as

$$\frac{ d{\boldsymbol{y}} }{ d{t} } = \boldsymbol{f}(t, \boldsymbol{y}) \quad, \quad \boldsymbol{y} \in \mathbb{R}^N$$ (4.7)

with the initial condition

$$\boldsymbol{y}(t_0) = \boldsymbol{y}_0 \quad.$$ (4.8)

The PVODE package [48,49] is a general purpose solver for initial-value problems for stiff and non-stiff ODEs of the form of Eq. (4.7). It is based on CVODE [50,51] and uses MPI for parallelization and portability.

Two methods are available for the numerical solution of Eq. (4.7): The backward differentiation formula (BDF), which is recommended for stiff problems, and the Adams-Moulton formula for non-stiff problems, both of which feature a variable stepsize and variable order. Both formulas can be written as the linear multistep formula

$$\sum_{i=0}^{K_1} \alpha_{n,i} y_{n-i} + h_n \sum_{i=0}^{K_2} \beta_{n,i} \frac{ d{y} _{n-i}}{ d{t} }=0 \quad,$$ (4.9)

where $h_n=t_n-t_{n-1}$ is the stepsize and $q$ the order. The Adams-Moulton formula is obtained with $K_1=1$ and $K_2=q-1$ with $1 \le q \le 12$. The BDF formula is represented by Eq. (4.9) with $K_1=q$ and $K_2=0$ with $1 \le q \le 5$. The numerical integration is started with $q=1$ and then varied automatically and dynamically.

If we insert Eq. (4.7) in Eq. (4.9) we get an implicit nonlinear system of equations for $\boldsymbol{y}$

$$\boldsymbol{G}(\boldsymbol{y}_n):=\boldsymbol{y}_n-h_n\beta... ...ldsymbol{f}(t_n,\boldsymbol{y}_n) - \boldsymbol{a}_n=0 \quad,$$ (4.10)

which has to be solved at each time step. $\beta_{n,0}$ and $\boldsymbol{a}_n$ depend on the method, the integration order, and the previous time steps. An efficient method for nonstiff problems is functional iteration, because it does not require the solution of a linear system of equations. However, for stiff problems it is better solved by Newton iteration, which does involve the solution of a linear system of equations. PVODE uses a Krylov subspace method - the iterative scaled preconditioned generalized minimal residual method (SPGMR) [52], whose performance can be considerably improved with suitable preconditioners.